Local unitary transformation, longrange quantum entanglement, wave function renormalization, and topological order
Abstract
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of longrange entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixedpoint wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the stringnet classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensorproduct wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetrybreaking orders in a generic tensorproduct state.
 Publication:

Physical Review B
 Pub Date:
 October 2010
 DOI:
 10.1103/PhysRevB.82.155138
 arXiv:
 arXiv:1004.3835
 Bibcode:
 2010PhRvB..82o5138C
 Keywords:

 64.70.Tg;
 71.27.+a;
 Quantum phase transitions;
 Strongly correlated electron systems;
 heavy fermions;
 Condensed Matter  Strongly Correlated Electrons;
 Quantum Physics
 EPrint:
 29 pages, 68 figures. Version 2, fixed point conditions refined, references added